quarterfinal
1
$$\int_1^{2022}\frac{\left\{x\right\}}{x}dx$$
2
$$\lim_{n\to\infty}n\int_0^{\frac{\pi}{4}}\tan^n{(x)}dx$$
3
$$\int_0^{\infty}\frac{x^{1010}}{(1+x)^{2022}}dx$$
4
$$\int\arcsin{(x)}\text{arccos}\,(x)dx$$
5
$$\max_{\{x_1, x_2, x_3, x_4, x_5, x_6, x_7\}=\{1, 2, 3, 4,5, 6, 7\}}\int_{\int_{x^1}^{x_2}x_3dx}^{\int_{x_4}^{x_5}x_6dx}x_7dx$$
6
$$\lim_{n\to\infty}\sqrt[n]{\int_0^2(1+6x-7x^2+4x^3-x^4)^ndx}$$
7
$$\int_0^1\frac{x^4}{\sqrt{1-x}}dx$$
8
$$\int_0^{\infty}\left\lceil \frac{1}{\left\lceil x \right\rceil-x} \right\rceil ^{-\left\lceil x \right\rceil}dx$$
9
$$\lim_{n\to\infty}n\int_0^{\infty}\sin{\Big(\frac{1}{x^n}\Big)}dx$$
10
$$\int_0^1\frac{-x+\sqrt{4-3x^2}}{2}dx$$
11
$$\lim_{n\to\infty}\sqrt{n}\int_{-\frac{1}{2}}^{\frac{1}{2}}(1-3x^2+x^4)^ndx$$
12
$$\int_{\frac{1}{2022}}^{2022}\frac{1+x^2}{x^2+x^{2022}}dx$$
semifinal
1
$$\int_0^{\infty}\frac{x(e^{-x}+1)}{e^x-1}dx$$
2
$$\int_0^{\infty}x^5e^{-x}\sin{x}\,dx$$
3
$$\int_{\frac{1}{2}}^2\ln{\biggl(\frac{\ln{(x+\frac{1}{x})}}{\ln{(x^2-x+\frac{17}{4})}}\biggl)}dx$$
4
$$\int_2^{\frac{5}{2}}\frac{(x^3-3x)^3-3(x^3-3x)}{\sqrt{x^2-4}}dx$$
5
$$\int_{-1}^1\biggl|\biggl|\biggl|\biggl||x|-\frac{2}{3}\biggl|-\frac{2}{3^2}\biggl|-\frac{2}{3^3}\biggl|-\dots\biggl|dx$$
6
$$\int\frac{1}{(x^2+1)^3}dx$$
7
$$\int_{-\infty}^{\infty}\frac{dx}{x^2-2x\cot{(x)}+\csc^2{(x)}}$$
8
$$\int_0^{\frac{\pi}{6}}\ln{(\sqrt3+\tan{x})}dx$$
final
1
$$\int\sqrt{(\sin{20x}+3\sin{21x}+\sin{22x})^2+(\cos{20x+3\cos{21x}+\cos{22x}})^2}dx$$
2
$$\int_0^{\infty}\frac{e^{-2x}\sin{3x}}{x}dx$$
3
$$\int_0^{2\pi}\cos{(2022x)}\frac{\sin{(10050x)}}{\sin{(50x)}}\frac{\sin{(10251x)}}{\sin{(51x)}}dx$$
$$\begin{array}{|c}\sin{10251x}=\sin{201\times51x}\\ \sin{10050x}=\sin{201\times50x}\end{array}$$$$\frac{\sin{nZ}}{\sin{Z}}=\frac{e^{inZ}-e^{-inZ}}{e^{iZ}-e^{-iZ}}=\frac{e^{-inZ}}{e^{-iZ}}\frac{e^{i2nZ}-1}{e^{i2Z}-1}=e^{-i(n-1)Z}\frac{(e^{2iZ})^n-1}{e^{2iZ}-1}$$$$\frac{\sin{nZ}}{\sin{Z}}=e^{-i(n-1)Z}\biggl(\sum_{k=0}^{n-1}e^{2ikZ}\biggl)$$$$\frac{\sin{(201Z)}}{\sin{(Z)}}=e^{-200iZ}\biggl(\sum_{k=0}^{200}e^{2ikZ}\biggl):$$$$\frac{\sin{(10050x)}}{\sin{(50x)}}\frac{\sin{(10251x)}}{\sin{(51x)}}=e^{-10000ix}\biggl(\sum_{k=0}^{200}e^{100ikx}\biggl)e^{-10200ix}\biggl(\sum_{k=0}^{200}e^{102ilx}\biggl)=e^{-20200ix}\biggl(\sum_{0\leqslant k, l\leqslant 200}e^{(100k+102l)ix}\biggl)$$$$cos{(2022x)}=e^{2022ix}:cos{(2022x)}\frac{\sin{(10050x)}}{\sin{(50x)}}\frac{\sin{(10251x)}}{\sin{(51x)}}=e^{-18178ix}\biggl(\sum_{0\leqslant k, l\leqslant 200}e^{(100k+102l)ix}\biggl)=$$$$=\sum_{0\leqslant k, l\leqslant 200}e^{(50k+51l-9089)2ix}$$$$I=\int_0^{2\pi}\sum_{0\leqslant k, l\leqslant 200}e^{(50k+51l-9089)2ix}dx$$$$\text{Note that }\int_0^{2\pi}e^{2inx}dx=\begin{equation}\begin{cases}2\pi, \text{ if }n=0\\0, \text{ if }n\ne 0\end{cases}\end{equation}$$$$\text{When n=0: }50k+51l=9089\rightarrow51l=9089[50]\rightarrow51l=39[50]\rightarrow l=50m+39\,\,(m\geqslant 0)$$$$50k+51(50m+39)=9089\Rightarrow k+51m=142\Rightarrow m\le \frac{142}{51}\Rightarrow 0\le m\le 2$$$$(k, l)=(50m+39, 142-51m) \text{ for m = 0,1,2}\Rightarrow n=0\text{ in 3 cases and }I=3\times2\pi=6\pi$$
Round 1
1
$$\int_0^{\frac{\pi}{2}}\frac{dx}{\cot{x}+\tan{x}}$$
2
$$\int_{-3}^{5}\sqrt{|x|^3}dx$$
3
$$\int\frac{e^{\frac{x}{2}}}{1+e^x}dx$$
4
$$\int\cot{x}\sec^2{x}dx$$
5
$$\int x^2(x^4-7)^2dx$$
6
$$\int_0^1(1+x+1+11+x+11x-x-111x-1010x)dx$$
7
$$\int_{-1}^2\Big|x^2-\frac{1}{4}\Big|dx$$
8
$$\int_1^2(x+1)^2\biggl(1-\frac{1}{x}\biggl)^3dx$$
9
$$\int_a^{3a}\biggl(\sqrt{\frac{a}{x}}+\sqrt{\frac{x}{a}}\biggl)^2dx$$
10
$$\int \biggl(\ln{x}+\frac{1}{x}\biggl)e^xdx$$
11
$$\int\frac{3x+5}{x^2+1}dx$$
12
$$\int_0^{\pi}\frac{\cos^2{x}}{1+\sin{x}}dx$$
13
$$\int\frac{x}{x^4-1}dx$$
14
$$\int e^{\sqrt{x}}dx$$
15
$$\int_{\frac{1}{2}}^2\biggl(\frac{\sin{\ln{x}}+\ln{x}}{x^3}\biggl)^{\frac{1}{3}}dx$$
16
$$\int\frac{dx}{x^2-2013x-4030}$$
17
$$\int\tan^3{x}\,dx$$
18
$$\int\Big(\frac{7}{10}x^5+\frac{3}{4}x^2-\frac{5}{19}x^3-\frac{31}{7}x^5\Big)dx$$
19
$$\int_0^4\sqrt{x}\sqrt{x\sqrt{x}+1}dx$$
20
$$\int_0^{\pi}\Big[\cos{\Big(x+\frac{\pi}{4}\Big)+\cos{\Big(x-\frac{\pi}{4}\Big)}}+\sin{4x}-\sin{\frac{x}{4}}\Big]dx$$
21
$$\int\sec{x}\cosh{x}(\cosh{x}\tan{x}+2\sinh{x})dx$$
Round 2
1
$$\int\frac{dx}{x\sqrt{x^4-1}}$$
2
$$\int_0^1\ln{(x^3+6x^2+11x+6)}dx$$
3
$$\int_0^{2015}\sin{\biggl(\frac{\pi}{2}\left\lfloor x \right\rfloor\biggl)}dx$$
4
$$\int_0^{\infty}\frac{\arctan{x}-\arctan{2x}}{x}dx$$
5
$$\int_0^{\infty}\frac{x}{x^3+1}dx$$
6
$$\int\sin{2x}\cos{3x}dx$$
7
$$\int_1^{\infty}a^{-\left\lfloor \log_b{x} \right\rfloor}dx$$
8
$$\int\frac{\sin{(\ln{(\sin{(\ln{x})}}))}\cos{(\ln{x}})}{x\sin{(\ln{x})}}dx$$
9
$$\int_0^{\frac{\pi}{2}}\ln{(\tan{x})}dx$$
10
$$\int\frac{e^{2x}+e^{3x}}{\cosh{x}}dx$$
11
$$\int\frac{\sin^2{x}}{\cos{(\frac{x}{2})}}dx$$
12
$$\int\Big(x\ln{x}-\frac{1}{x}\Big)e^xdx$$
13
$$\int e^x\sin{x}\cos^2{x}dx$$
14
$$\int x^{2015}e^{x^{1008}}dx$$
15
$$\int_0^{\frac{\pi}{16}}x\sec^2{4x}dx$$
16
$$\int\frac{\sin{x}}{1+\sin{x}}dx$$
Round3
1
$$\int_0^{\infty}\frac{1}{(1+x^{2015})(1+x^2)}dx$$
2
$$\int_0^{\infty}(xe^{1-x}-\left\lfloor x \right\rfloor e^{1-\left\lfloor x \right\rfloor})dx$$
3
$$\int\frac{2000x^{2014}+14}{x^{2015}-x}dx$$
4
$$\int_0^{\frac{\pi}{4}}x\prod_{k=1}^{\infty}\cos{\Big(\frac{x}{2^k}\Big)}dx$$
5
$$\int_0^{\frac{\pi}{4}}\frac{1-x^2+x^4-x^6+x^8-x^{10}+\dots}{\cos^2{x}+\cos^4{x}+\cos^6{x}+\dots}dx$$